3.1326 \(\int (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=274 \[ \frac{d^{7/2} \left (b^2-4 a c\right )^{17/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{462 c^3 \sqrt{a+b x+c x^2}}+\frac{d^3 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{231 c^2}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{9/2}}{110 c^2 d}+\frac{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}}{385 c^2}+\frac{\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{9/2}}{15 c d} \]

[Out]

((b^2 - 4*a*c)^3*d^3*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(231*c^2) + ((b^
2 - 4*a*c)^2*d*(b*d + 2*c*d*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(385*c^2) - ((b^2 -
4*a*c)*(b*d + 2*c*d*x)^(9/2)*Sqrt[a + b*x + c*x^2])/(110*c^2*d) + ((b*d + 2*c*d*
x)^(9/2)*(a + b*x + c*x^2)^(3/2))/(15*c*d) + ((b^2 - 4*a*c)^(17/4)*d^(7/2)*Sqrt[
-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x]/((b
^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(462*c^3*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.648472, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{d^{7/2} \left (b^2-4 a c\right )^{17/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{462 c^3 \sqrt{a+b x+c x^2}}+\frac{d^3 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{231 c^2}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{9/2}}{110 c^2 d}+\frac{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2}}{385 c^2}+\frac{\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{9/2}}{15 c d} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(3/2),x]

[Out]

((b^2 - 4*a*c)^3*d^3*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(231*c^2) + ((b^
2 - 4*a*c)^2*d*(b*d + 2*c*d*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(385*c^2) - ((b^2 -
4*a*c)*(b*d + 2*c*d*x)^(9/2)*Sqrt[a + b*x + c*x^2])/(110*c^2*d) + ((b*d + 2*c*d*
x)^(9/2)*(a + b*x + c*x^2)^(3/2))/(15*c*d) + ((b^2 - 4*a*c)^(17/4)*d^(7/2)*Sqrt[
-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x]/((b
^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(462*c^3*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 131.032, size = 258, normalized size = 0.94 \[ \frac{\left (b d + 2 c d x\right )^{\frac{9}{2}} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{15 c d} + \frac{d^{3} \left (- 4 a c + b^{2}\right )^{3} \sqrt{b d + 2 c d x} \sqrt{a + b x + c x^{2}}}{231 c^{2}} + \frac{d \left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{5}{2}} \sqrt{a + b x + c x^{2}}}{385 c^{2}} - \frac{\left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{9}{2}} \sqrt{a + b x + c x^{2}}}{110 c^{2} d} + \frac{d^{\frac{7}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{17}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{462 c^{3} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**(7/2)*(c*x**2+b*x+a)**(3/2),x)

[Out]

(b*d + 2*c*d*x)**(9/2)*(a + b*x + c*x**2)**(3/2)/(15*c*d) + d**3*(-4*a*c + b**2)
**3*sqrt(b*d + 2*c*d*x)*sqrt(a + b*x + c*x**2)/(231*c**2) + d*(-4*a*c + b**2)**2
*(b*d + 2*c*d*x)**(5/2)*sqrt(a + b*x + c*x**2)/(385*c**2) - (-4*a*c + b**2)*(b*d
+ 2*c*d*x)**(9/2)*sqrt(a + b*x + c*x**2)/(110*c**2*d) + d**(7/2)*sqrt(c*(a + b*
x + c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(17/4)*elliptic_f(asin(sqrt(b*d + 2
*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(462*c**3*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 2.10694, size = 295, normalized size = 1.08 \[ \frac{(d (b+2 c x))^{7/2} \left (\frac{c (a+x (b+c x)) \left (16 b^2 c^2 \left (36 a^2+345 a c x^2+518 c^2 x^4\right )+32 b c^3 x \left (12 a^2+238 a c x^2+231 c^2 x^4\right )+32 c^3 \left (-20 a^3+12 a^2 c x^2+119 a c^2 x^4+77 c^3 x^6\right )+2 b^4 c \left (35 a+453 c x^2\right )+16 b^3 c^2 x \left (107 a+266 c x^2\right )-5 b^6+10 b^5 c x\right )}{(b+2 c x)^3}+\frac{5 i \left (b^2-4 a c\right )^4 \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{\sqrt{-\sqrt{b^2-4 a c}} (b+2 c x)^{5/2}}\right )}{2310 c^3 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(3/2),x]

[Out]

((d*(b + 2*c*x))^(7/2)*((c*(a + x*(b + c*x))*(-5*b^6 + 10*b^5*c*x + 16*b^3*c^2*x
*(107*a + 266*c*x^2) + 2*b^4*c*(35*a + 453*c*x^2) + 32*b*c^3*x*(12*a^2 + 238*a*c
*x^2 + 231*c^2*x^4) + 16*b^2*c^2*(36*a^2 + 345*a*c*x^2 + 518*c^2*x^4) + 32*c^3*(
-20*a^3 + 12*a^2*c*x^2 + 119*a*c^2*x^4 + 77*c^3*x^6)))/(b + 2*c*x)^3 + ((5*I)*(b
^2 - 4*a*c)^4*Sqrt[(c*(a + x*(b + c*x)))/(b + 2*c*x)^2]*EllipticF[I*ArcSinh[Sqrt
[-Sqrt[b^2 - 4*a*c]]/Sqrt[b + 2*c*x]], -1])/(Sqrt[-Sqrt[b^2 - 4*a*c]]*(b + 2*c*x
)^(5/2))))/(2310*c^3*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.07, size = 1057, normalized size = 3.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^(7/2)*(c*x^2+b*x+a)^(3/2),x)

[Out]

1/4620*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d^3*(-1280*a^4*b*c^4+1152*a^3*b^3
*c^3+140*a^2*b^5*c^2-10*a*b^7*c+5*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/
2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(
-4*a*c+b^2)^(1/2))^(1/2)*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2
)^(1/2))^(1/2)*2^(1/2),2^(1/2))*(-4*a*c+b^2)^(1/2)*b^8+1280*((b+2*c*x+(-4*a*c+b^
2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2
*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticF(1/2*((b+2*c*x+(-4*a
*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*(-4*a*c+b^2)^(1/2)*a^4
*c^4+87808*x^6*a*b*c^7+123328*x^5*a*b^2*c^6+41920*x^4*a^2*b*c^6+88800*x^4*a*b^3*
c^5+42688*x^3*a^2*b^2*c^5+33728*x^3*a*b^4*c^4-1536*x^2*a^3*b*c^5+22112*x^2*a^2*b
^3*c^4+5696*x^2*a*b^5*c^3+1792*x*a^3*b^2*c^4+4856*x*a^2*b^4*c^3+140*x*a*b^6*c^2+
9856*x^9*c^9-80*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x
+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))
^(1/2)*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(
1/2),2^(1/2))*(-4*a*c+b^2)^(1/2)*a*b^6*c+44352*x^8*b*c^8-1280*((b+2*c*x+(-4*a*c+
b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b
-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticF(1/2*((b+2*c*x+(-4
*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*(-4*a*c+b^2)^(1/2)*a
^3*b^2*c^3+25088*x^7*a*c^8+82432*x^7*b^2*c^7+81536*x^6*b^3*c^6+16768*x^5*a^2*c^7
+45736*x^5*b^4*c^5+13988*x^4*b^5*c^4-1024*x^3*a^3*c^6+1852*x^3*b^6*c^3-10*x^2*b^
7*c^2-2560*x*a^4*c^5-10*x*b^8*c+480*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(
1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))
/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b
^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*(-4*a*c+b^2)^(1/2)*a^2*b^4*c^2)/c^3/(2*c^2*x^3
+3*b*c*x^2+2*a*c*x+b^2*x+a*b)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")

[Out]

integrate((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (8 \, c^{4} d^{3} x^{5} + 20 \, b c^{3} d^{3} x^{4} + a b^{3} d^{3} + 2 \,{\left (9 \, b^{2} c^{2} + 4 \, a c^{3}\right )} d^{3} x^{3} +{\left (7 \, b^{3} c + 12 \, a b c^{2}\right )} d^{3} x^{2} +{\left (b^{4} + 6 \, a b^{2} c\right )} d^{3} x\right )} \sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)^(3/2),x, algorithm="fricas")

[Out]

integral((8*c^4*d^3*x^5 + 20*b*c^3*d^3*x^4 + a*b^3*d^3 + 2*(9*b^2*c^2 + 4*a*c^3)
*d^3*x^3 + (7*b^3*c + 12*a*b*c^2)*d^3*x^2 + (b^4 + 6*a*b^2*c)*d^3*x)*sqrt(2*c*d*
x + b*d)*sqrt(c*x^2 + b*x + a), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x+b*d)**(7/2)*(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.84457, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)^(3/2),x, algorithm="giac")

[Out]

Done